Interest

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In finance, interest is the price paid by a borrower for the use of a lender's money. In other words, interest is the amount of paid to "rent" money for a period of time. The original amount lent is called the principal, and the percentage of the principal which must be paid annually as interest is called the interest rate. Interest rates are crucial indicators in financial markets.

The existence of interest

There are a variety of reasons which explain the why lenders charge interest for the use of their money:

The time value of money: Most people would choose to have money in the present rather than money in the future. When asked to lend their current money in exchange for a promise to repay that money in the future, most lenders will agree only if they are repaid more than they originally lent. In effect, the interest rate is the payment for the use of money over time.

Alternative investments. The lender has a choice between using his money in different investments. If he chooses one, he forgoes the returns from all the others. In other words, lending incurrs an opportunity cost due to the possible alternative uses of the lent money.

Inflationary expectations. Most economies generally exhibit inflation, meaning a given amount of money buys fewer goods in the future than it will now. The borrower needs to compensate the lender for this.

Risks of investment. There is always a risk that the borrower will go bankrupt, abscond, or otherwise default on the loan. This means that a lender generally charges a risk premium to ensure that, across his investments, he is compensated for those that fail.

Liquidity preference. People prefer to have their resources available in a form that can immediately be exchanged, rather than a form that takes time or money to realise.

Taxes. Because some of the gains from interest may be subject to taxes, the lender may insist on a higher rate to make up for this loss.

History

Historical documents dating back to the Sumerian civilization, circa 3000 B.C., reveal that the ancient world had developed a formalized system of credit based on two major commodities, grain and silver. Before there were coins, metal loans were based on weight. Archaeologists have uncovered pieces of metal that were used in trade in Troy, Minoan and Mycenaean civilizations, Babylonia, Assyria, Egypt and Persia. Before money loans came into existence, loans of grain and silver served to facilitate trade. Silver was used in town economies, while grain was used in the country.

The collection of interest was restricted by Jewish, Christian and other religions under laws of usury (essentially a derogatory term for interest). This is still the case with Islam, which mandates no-interest Islamic finance.

Irving Fisher is largely responsible for shaping the modern concept of interest with his 1930 work, The Theory of Interest.

Types of accrual

Most interest accrues (accumulates) as either simple interest or compound interest.

Simple interest

Simple interest is interest that accrues linearly. In other words, it grows by a certain fraction of the principal per time period. Calculation of accrued interest of most debt uses simple interest. Once an interest payment is made, the lender can reinvest it elsewhere. In case they reinvest it in the original investment, interest will start accruing on this interest. In this case, they can calculate the growth of their investment using the compound interest method.

$A(t) = A_0 \cdot (1 + t \cdot r)\,$

$A(t)$ = Amount after $t$ years
$A_0$ = Principal (start amount)
$r$ = Interest rate
$t$ = Time in years

(Note: The interest rate must be entered as a decimal (such as 0.06) rather than a percentage (such as 6%).

A(t) is called the amount function. The constant coefficient $A_0$ is usually dropped in mathematics of interest calculation, and the resulting accumulation function is used instead:

a(t)=1+t  \cdot r\,

Compound interest

Compound interest, previously called anatocism, is interest which is regularly added to the debt (compounded). Interest is then calculated not only over the principal, but also over the interest that has been added to the debt before--in other words, it is calculated over the total amount owed. With compound interest, the frequency of compounding influences the total amount of interest paid over the life of the loan. The amount function for compound interest is an exponential function in terms of time.

$A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t}$

$n$ = Number of compounding periods per year (note that the total number of compounding periods is $n \cdot t$ )

As $n$ increases the rate approaches an upper limit of $e ^ r$ . This rate is called continuous compounding.

Many banks advertise an annual percentage yield (APY) which is the return on the principal over an entire year. For example, a 5% rate compounded monthly would have an approximate APY of 5.12%.

Rule of 78

Some consumer loans calculate interest by the "Rule of 78" or "Sum of digits" method. Seventy-eight is the sum of the numbers 1 through 12, inclusive. And the practice enabled quick calculations of interest in the pre-computer days. In a loan with interest calculated per the Rule of 78, the total interest over the life of the loan is calculated as either simple or compound interest and amounts to the same as either of the above methods. Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78 is to make early pay-offs of term loans more expensive. Approximately 3/4 of all interest due on a one year loan is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than than the APY used to calculate the payments. ^*

The United States outlawed the use of "Rule of 78" interest in loans over five years in term. Certain other jurisdictions have outlawed application of the Rule of 78 in certain types of loans, particularly consumer loans. ^*

Types of interest rate

Interest rates can be divided into two types:

Fixed. The interest rate stays fixed throughout the life of the debt. Most bonds are fixed rate bonds.
Variable. The interest rate is usually determined by a reference rate, such as LIBOR or a consumer price index. Examples are floating rate notes. Sometimes interest rates depend directly on financial ratios or the credit rating of the borrowing company.

It is common for firms to swap between the two types of interest rate. These contractual agreements are derivatives called interest rate swaps. GAAP provides guidelines for some of these kinds of changes.

Analysis of interest-rate risks

Interest involves the future, which is uncertain. Some interest bearing investments are riskier than others are. The greater the risk of the security, the more interest the investors will expect to receive.

The fundamental determinants of interest rate of a debt instrument are these risks. The following is a list of risks commonly associated with interest rates:

Nonsystematic risks
- Credit risk – the risk of default on the loan or of bankruptcy
- Maturity/Term risk – the risk involved in a long-term investment
- Liquidity risk – the need of compensating the illiquidity of the debt
Systematic risks
- Inflation risk – macroeconomic price changes
- Exchange rate risk – currency fluctuation

Credit risk

The credit risk is the most commonly associated risk. It determines the different amount individuals or firms pay based on their credit-worthiness. Different parties will be offered different rates on debt obligations (such as loans). The measure of credit worthiness of an individual is called a credit rating or credit score. Other entities (such as governments and companies) will acquire a bond rating if they are active in bond markets.

The credit spread between an instrument and its risk-free equivalent is called the risk premium.

Maturity/term risk

See term structure of interest rates

Liquidity risk

Liquidity risk is the risk that the lender might not be able to liquidate the debt on short notice. The difference in interest rate due to liquidity risk is called liquidity spread. Instruments such as bonds have an active secondary market. Other instruments such as savings deposits are easily transferable to cash. On the other hand 30-year US Government savings bond is non-transferable. It can only be redeemed at half price before maturity. The savings bond will obviously offer a higher return.

Another interesting phenomenon observed from liquidity spread is that on-the-run securities (primary market) have lower interest rates compare to the off-the-run securities (secondary market). This implies that there is a higher demand for on-the-run securities.

Inflation and exchange-rate risks

The largest inflation and exchange rate risks in the industrialized world come from loans to developing countries. Therefore, loans offered by banks in developed countries usually denominate the loan contract in stable currencies such as the US Dollar, Pound Sterling, or Euro.

This has led to unfavorable consequences for the borrowers of developing countries because the economies of developing countries often have high inflation and an unstable exchange rate.

During times of especially high inflation in developed nations (e.g. the late 1970s in the United States) many banks suffered significant losses in attempts to use currency trading to offset the higher domestic inflation rates.

External links

Online Compound Interest Calculator

Basic financial concepts | Mathematical finance

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